

504K 

This movie shows the twodimensional shadow of a cube as it rotates. Two
of the faces are colored to help you follow how the shadow changes as the
cube turns in space. At the start of the movie, the red square is closest
to the light source (or viewer), and so it appears larger, while the blue
face is farther away, and seems smaller. As the cube rotates, the blue
square seems to shift to the side and both squares appear to become distorted.
On the cube itself, they are still squares, but in the twodimensional
shadow, their images are trapezoids. As the cube rotates further, the blue
square "pops" through the side of the red square, and eventually is
completely outside it. The red square flattens out and "turns inside out"
as we move from seeing one side of this face of the cube to seeing the other
side. In a view of the cube where the red and blue faces are equally
sized trapezoids, we see that these two faces are now the sides of the
cube. As the cube turns further, the blue faces comes to the front, and is
the largest square, while the red one moves to the back, and becomes
smaller: the red and blue have interchanged positions. The rotation
continues through the sequence again until the colored faces are back at
the their starting positions.
We can easily reconstruct in our minds the threedimensional cube from
these twodimensional pictures. The hard part is actually thinking of
them as flat images! But this is what we have to do in order to make
sense of the next movie, which shows the threedimensional shadows of a
hypercube rotating in four dimensions.



660K 

This movie shows the analogous threedimensional shadows of the
fourdimensional hypercube. Our initial view is of a large red cube
containing a smaller blue one (some faces are removed to make it easier to
see the interior structure). The red cube is closer to the light source
(or the viewer), and so has a larger shadow. As the hypercube begins to
rotate, the blue face moves to the side and both begin to become distorted
as one side moves closer to the light and another farther away. At some
point, the blue face passes through the side of the red one, and
eventually pulls all the way through. The red cube begins to flatten out
(as it becomes parallel to the viewing direction), and then appears to turn inside
out, as the fourdimensional view moves from one side of the cube to othe
other. At one point, the red and blue cubes form a symmetric view; each cube
is now one of the sides of the hypercube and two different cubes are at
the front and back. The rotation continues for another 90 degrees until the
blue cube is at the front (largest) and the red one at the back
(smallest). The sequence repeats itself with the colors interchanged as
the blue cube moves again to the back. We end where we began, with a
large red cube containing a small blue one.
